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Full text of "Men Of Mathematics"

THE COPERNICUS  OF GEOMETRY
equal, but, without using this postulate, it is impossible to prove
that AXY, BYX are right angles, although they look it. If we
assume the parallel postulate we can prove that AXY, BYX are
right angles and, conversely, if we assume that AXY, BYX are
right angles, we can prove the parallel postulate. Thus the
assumption that AXY, BYX are right angles is equivalent to the
parallel postulate. This assumption is to-day called the hypothesis
of the right angle (since both angles are right angles the singular
instead of the plural 'angles' is used).
It is known that the hypothesis of the right angle leads to a
consistent, practically useful geometry, in fact to Euclid's
geometry refurbished to meet modern standards of logical
rigour. But the figure suggests two other possibilities: each of
the equal angles AXY9 BYX is less than a right angle - the
hypothesis of the acute angle; each of the equal angles AXY,
BYX is greater than a right angle - the hypotJiesis of the obtuse
angle. Since any angle can satisfy one, and only one, of the
requirements that it be equal to, less than, or greater than a right
angle, the three hypotheses - of the right angle, acute angle,
and obtuse angle respectively - exhaust the possibilities.
Common experience predisposes us in favour of the first
hypothesis. To see that each of the others is not as unreasonable
as might at first appear we shall consider something closer to
actual human experience than the highly idealized 'plane' in
which Euclid imagined his figures drawn. But first we observe
that neither the hypothesis of the acute angle nor that of the
obtuse angle will enable us to prove Euclid's parallel postulate,
because, as has been stated above, Euclid's postulate is equi-
valent to the hypothesis of the right angle (in the sense of inter-
deducibility ; the hypothesis of the right angle is both necessary
and sufficient for the deduction of the parallel postulate)*
Hence if we succeed in constructing geometries on either of the
two new hypotheses, we shall not find in them parallels in
Euclid's sense.
To make the other hypotheses less unreasonable than they
may seem at first sight, suppose the Earth were a perfect sphere
(without irregularities due to mountains, etc.). A plane drawn
through the centre of this ideal Earth cuts the surface in a great